Scaling and bounds in thermal conductivity of planar Gaussian correlated microstructures

In this study 2d two phase microstructures closely resembling the experimentally captured micrographs of the interpenetrating phase composites are generated using a Gaussian correlation function based method. The scale dependent bounds on the effective thermal conductivity of such microstructures are then studied using Hill-Mandel boundary conditions. A scaling function is formulated to describe the transition from statistical volume element (SVE) to representative volume element (RVE), as a function of the mesoscale δ, the correlation length of the Gaussian correlation function λ, the volume fraction v, and the contrast k between the phases. The scaling function is determined through fitting the data from extensive simulations conducted over the parameter space. The scaling function shows that SVE approaches RVE as $(δ/λ)−1.16$⁠. A material scaling diagram allows estimation of the RVE size, to within a chosen accuracy, of a given microstructure characterized by the correlation length of the Gaussian correlation function, contrast, and volume fraction of the phases.